• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.725-737
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• 1995

Let $X_t = (X_t^(1), X_t^(2))', t \geqslant 0$, be a real stationary 2-dimensional Gaussian process with $EX_t^(1) = EX_t^(2) = 0$ and $$ EX_0 X'_t = (_{\rho(t) r(t)}^{r(t) \rho(t)}), $$ where $r(t) \sim $\mid$t$\mid$^-\alpha, 0 < \alpha < 1/2, \rho(t) = o(r(t)) as t \to \infty, r(0) = 1, and \rho(0) = \rho (0 \leqslant \rho < 1)$. For $t > 0, u > 0, and \upsilon > 0, let L_t (u, \upsilon)$ be the time spent by $X_s, 0 \leqslant s \leqslant t$, above the level $(u, \upsilon)$.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1141-1158
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• 2018

Hahn difference operator $D_{q,{\omega}}$ which is defined by $$D_{q,{\omega}}g(t)=\{{\frac{g(gt+{\omega})-g(t)}{t(g-1)+{\omega}}},{\hfill{20}}\text{if }t{\neq}{\theta}:={\frac{\omega}{1-q}},\\g^{\prime}({\theta}),{\hfill{83}}\text{if }t={\theta}$$ received a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the form $$D_{q,{\omega}}x(t)=A(t)x(t)+f(t),\;t{\in}I$$, and $$D^2{q,{\omega}}x(t)+A(t)D_{q,{\omega}}x(t)+R(t)x(t)=f(t),\;t{\in}I$$, where $A,R:I{\rightarrow}{\mathbb{X}}$, and $f:I{\rightarrow}{\mathbb{X}}$. Here ${\mathbb{X}}$ is a Banach algebra with a unit element e and I is an interval of ${\mathbb{R}}$ containing ${\theta}$.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.227-232
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• 2006

We study some dynamical properties in the context of bitransformation groups, and show that if (H,X,T) is a bitransformation group such that (H,X) is almost periodic and (X/H,T) is pointwise almost periodic $T_2$ and $x{\in}X$, then $E_x=\{q{\in}E(H,X){\mid}qx{\in}{\overline{xT}\}$ is a compact $T_2$ topological group and $E_{qx}=E_x(q{\in}E(H,X))$ when H is abelian, where E(H,X) is the enveloping semigroup of the transformation group (H,X).

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.317-325
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• 1994

Suppose X is a complex Banach space and write B(X) for the Banach algebra of bounded linear operators on X, X＊ for the dual space of X, and T＊$\in$ B(X＊) for the dual operator of T. For T $\in$ B(X) write a(T) = dim T$^{-1}$ (0) and $\beta$(T) = codim T(X).(omitted)

• Honam Mathematical Journal
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• v.28 no.4
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• pp.591-603
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• 2006

For a set $X{\subset}{\mathbb{Z}}^n$ let $(X,\;T^n_X)$ be the subspace of the Khalimsky n-dimensional space $({\mathbb{Z}}^n,\;T^n)$, $n{\in}N$. Considering a k-adjacency of $(X,\;T^n_X)$, we use the notation $(X,\;k,\;T^n_X)$. In this paper for a map $$f:(X,\;k,\;T^n_X){\rightarrow}(Y,\;2\;T_Y)$$, we find the condition that weak (k, 2)-continuity of the map f implies strong (k, 2)-continuity of f.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.91-101
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• 2008

Let H be a Hilbert space. Assume that $0{\leq}{\alpha}$, ${\beta}{\leq}1$ and r(t) > 0 in I = [0, T]. By means of the fixed point theorem of Leray-Schauder type the existence principles of solutions for two point boundary value problems of the form $\array{(r(t)x^{\prime}(t))^{\prime}+f(t,x(t),r(t)x^{\prime}(t))=0,\;t{\in}I\\x(0)=x(T)=0}$ are established where f satisfies for positive constants a, b and c ${\mid}{f(t,x,y){\mid}{\leq}a{\mid}x{\mid}^{\alpha}+b{\mid}y{\mid}^{\beta}+c\;\;for\;all(t,x,y){\in}I{\times}H{\times}H$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.401-409
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• 2010

In this paper, we introduce an iterative method to study oscillatory properties of delay difference equations of the following form ${\nabla}_{\alpha}\;[x(t)\;-\;r(t)x(t\;-\;k)]\;+\;p(t)x(t\;-\;{\tau})\;-\;q(t)x(t\;-\;{\sigma})\;=\;0$, $t\;{\geq}\;t_0$, where $t_0\;{\in}\;\mathbb{R}$, t varies in the real interval ($t_0,\;{\infty}$), $\alpha$ > 0, $\kappa$, $\tau$, ${\sigma}\;{\geq}\;0$, $r\;{\in}\;C\;([t_0-{\alpha},\;{\infty}),\;\mathbb{R}^+$, p, $q\;{\in}\;C\;([t_0,\;{\infty}),\;\mathbb{R}^+)$ and ${\nabla}_{\alpha}x(t)\;=\;x(t)\;-\;x(t\;-\;{\alpha})$ for $t\;{\geq}\;t_0$.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.581-589
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• 2009

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term and forcing term: $$(r(t)k_1(x(t),x'(t)))'+p(t)k_2(x(t),x'(t))x'(t)+q(t)f(x(t))=0$$. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity. And, as a consequence, our results apply to wider classes of nonlinear differential equations. Some illustrative examples are considered.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.637-650
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• 2008

New sufficient conditions for the existence of at least one solution of Neumann type boundary value problems for second order nonlinear differential equations $$\array{\{{p(t)\phi(x'(t)))'=f(t,x(t),\;x(\tau_1(t)),\;{\cdots},\;x(\tau_m(t))),\;t\in[0,T],\\x'(0)=0,\;x'(T)=0,}\,}$$, are established.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.217-227
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• 2016

Let X be a nonempty set, and let $\mathfrak{F}=\{Y_i:i{\in}I\}$ be a family of nonempty subsets of X with the properties that $X={\bigcup}_{i{\in}I}Y_i$, and $Y_i{\cap}Y_j={\emptyset}$ for all $i,j{\in}I$ with $i{\neq}j$. Let ${\emptyset}{\neq}J{\subseteq}I$, and let $T^{(J)}_{\mathfrak{F}}(X)=\{{\alpha}{\in}T(X):{\forall}i{\in}I{\exists}_j{\in}J,Y_i{\alpha}{\subseteq}Y_j\}$. Then $T^{(J)}_{\mathfrak{F}}(X)$ is a subsemigroup of the semigroup $T(X,Y^{(J)})$ of functions on X having ranges contained in $Y^{(J)}$, where $Y^{(J)}:={\bigcup}_{i{\in}J}Y_i$. For each ${\alpha}{\in}T^{(J)}_{\mathfrak{F}}(X)$, let ${\chi}^{({\alpha})}:I{\rightarrow}J$ be defined by $i{\chi}^{({\alpha})}=j{\Leftrightarrow}Y_i{\alpha}{\subseteq}Y_j$. Next, we define two congruence relations ${\chi}$ and $\widetilde{\chi}$ on $T^{(J)}_{\mathfrak{F}}(X)$ as follows: $({\alpha},{\beta}){\in}{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}={\chi}^{({\beta})}$ and $({\alpha},{\beta}){\in}\widetilde{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}{\mid}_J={\chi}^{({\alpha})}{\mid}_J$. We begin this paper by studying the regularity of the quotient semigroups $T^{(J)}_{\mathfrak{F}}(X)/{\chi}$ and $T^{(J)}_{\mathfrak{F}}(X)/{\widetilde{\chi}}$, and the semigroup $T^{(J)}_{\mathfrak{F}}(X)$. For each ${\alpha}{\in}T_{\mathfrak{F}}(X):=T^{(I)}_{\mathfrak{F}}(X)$, we see that the equivalence class [${\alpha}$] of ${\alpha}$ under ${\chi}$ is a subsemigroup of $T_{\mathfrak{F}}(X)$ if and only if ${\chi}^{({\alpha})}$ is an idempotent element in the full transformation semigroup T(I). Let $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ be the sets of functions in $T_{\mathfrak{F}}(X)$ such that ${\chi}^{({\alpha})}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [${\alpha}$], $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ of $T_{\mathfrak{F}}(X)$.